168 ANALYTICAL MECHANICS and F cos e ds = (IX +mY + nZ} ds = Xdx + Ydy + Z dr, where dx, dy, and dz are the components of ds along the axes. Thus the total work done in a finite displacement is given by W = I Fcoseds Jo *dz. (II) fyYdy+ Jo Equation (II) states that the work done by a force equals the sum of the amounts of work done by its components. PROBLEMS. 1. Find the number of foot-pounds in one Joule. 2. Find the number of ergs in one foot-pound. 3. Find the work done in dragging a weight w up an inclined plane of length I, height h, and coefficient of friction JJL. 4. A body of 100 kg. mass is dragged up, then down, an inclined plane. 'Compare the work done in the two cases if the length of the plane is 15 m., .the height 5 'in., and the coefficient of friction 0.5. 5. What is the work done in winding a uniform chain which hangs from •a horizontal cylinder? The chain is 25 m. long, and has a mass of 125 kg. 6. A body has to be dragged from a point at the base of a conical hill to a point diametrically opposite. Show that, if the angle which the sides of the hill make with the horizon equals the angle of friction, the work done in dragging the body over the hill is less than in dragging it around .the base. .. 7. A steam hammer falls vertically from a height of 3 feet under the •action of its own weight and of a force of 2000 pounds due to steam pres-.sure. At the end of its fall it makes a dent of 1 inch depth in an iron plate. Find the total amount of work done in making the dent. The hammer weighs 1000 pound^. 8. In the preceding problem find the average resisting force. 9. A locomotive which is capable of exerting a draw-bar pull of 1.5 tons is coupled to a train of six cars. The locomotive and the tender weigh 50 tons. The cars weigh 15 tons each. Find the time it takes the locomotive to impart to the train a velocity of 60 miles per hour and the work done under the following conditions.